There is information on NURBS equations on wikipedia.
There is also information from Rhino on .3dm files, which store the NURBS...

Hi OSTexo,

> Is there some sort of common equation that defines a NURBS curve no matter what its "G" ness is?

Yeah there is, but it's kind of more like a mechanism that involves several different pieces of data which then together form the curve.

There's a "knot vector" which is a sequence of numbers, there is one formula that takes the relative spacing of these numbers and then yields "basis functions". The basis functions define how much of each control point for each span of the curve factors into the curve's evaluated position.

Control points can be made up of x, y, z, and also "weight" values. When the weight value is used, it makes the curve what's called a "rational" curve, where the x,y,z,w values form a kind of 4 dimensional curve and it's projected into 3D space by dividing each of x y and z by the weight value. This division part of the process allows for a conic section to result when certain weight values are used.

The whole process of these equations is more like a mechanism that describes how control points are averaged together as you move along the curve. It's not quite like the equations by themselves form a specific shape like I think you're kind of thinking about.


> If you would imagine a bowl shape that has a different set of inner and outer curves
> I'm trying to find out if a relationship or some ratio can be mathematically expressed
> between those two curve sets that are revolved to form the bowl.

There isn't anything inherent in the NURBS formulation itself that says anything about connections like this between inner and outer curves. If there is a connection between them, it's due to some higher level process that was guiding the positioning of all the control points of all the various separate pieces of geometry involved.

Reverse engineering something like that is probably going to involve evaluating points on the different pieces of geometry and having some process of comparing them with each other, it's not likely that some inherent equation in the NURBS formulation itself is going to do anything for this in particular.

- Michael

Hello Michael,

Thank you very much for the explanation. It sounds like the answer to the relationship is different depending on what points you choose to evaluate on each inner and outer curve pair. I'm actually good with that answer if I'm reading you correctly.

Hi OSTexo,

> It sounds like the answer to the relationship is different depending on what points you choose to
> evaluate on each inner and outer curve pair.

Yeah I suppose so. I guess you might want to define your comparison points by some kind of geometric searching function, like "the closest point to the other curve from this given point on the first curve".

Maybe an equation using that kind of geometric point to point solving function between 2 objects is more along the lines of what you're looking for. But there is nothing built into the inherent NURBS properties of curves individually that would give you an equation like that, you would need to do some pretty sophisticated reverse engineering process that involved doing going through the curve, running the point to point solving function to get a comparison point on the other curve and making note of how that distance changes as you move across the curve. But unless you already have some idea of what kind of relationship you're looking for it's not really very approachable to just say "find any equation that produces this point matching set".

- Michael